First, we set some notation and we
recall some results from [1]}.
Let \(n\in \mathbb {Z}\) , \(n\ge 1\) .
We denote \(n\) -uples of points in \(M\)  as \(\mathbf {a}= (a_1, \, \ldots , \, a_n)\) 
and \(n\) -uples of integer numbers as \(\mathbf {d}= (d_1, \, \ldots , \, d_n)\) .
We restrict our attention to the pairs \((\mathbf {a}, \, \mathbf {d})\)  in the `admissible class'
\( {A}^n := \left\lbrace (\mathbf {a}, \, \mathbf {d})\in M^n\times \mathbb {Z}^n\colon a_j \ne a_k \ \textrm { for any } j\ne k \ \textrm { and } \ \sum _{j=1}^n d_j = \chi (M)\right\rbrace \) 
